4 linked-list data structures linked-lists: singly-linked-lists, doubly-linked-lists insertion ...

4Linked-List Data Structures

Linked-lists: singly-linked-lists, doubly-linked-lists

Insertion

Deletion

Searching

© 2008 David A Watt, University of Glasgow

Algorithms & Data Structures (M)

4-2

Linked-lists (1)

A linked-list consists of a header together with a sequence of nodes connected by links:

– Each node (except the last) has a successor node.

– Each node (except the first) has a predecessor node.

– Each node contains a single element (value or object), plus links to its successor and/or predecessor.

ant bat cat

header null linklinknode element

ant bat cat

4-3

Linked-lists (2)

The length of a linked-list is its number of nodes.

An empty linked-list has no nodes.

We can manipulate a linked-list’s elements.

We can also manipulate a linked-list’s links, and thus change its very structure! E.g.:

ant bat catBefore:

ant bat catAfter:

whereas an array’s structure can’t be changed

4-4

Singly-linked-lists (1)

A singly-linked-list (SLL) consists of a header together with a sequence of nodes connected by links in one direction only:

– Each SLL node contains a single element, plus a link to the node’s successor (or null if the node has no successor).

– The SLL header contains a link to the SLL’s first node (or a null link if the SLL is empty).

pig dog ratcat

dog

empty SLL

4-5


Java class implementing SLLs:

public class SLL {// Each SLL object is the header of a // singly-linked-list.

private SLL.Node first;

public SLL () {// Construct an empty SLL.this.first = null;

}

… methods (to follow)

4-6


Java class implementing SLLs (continued):////////// Inner class //////////

private static class Node {// Each SLL.Node object is a node of a // singly-linked-list.

protected Object element;protected Node succ;

public Node (Object elem, Node succ) {

this.element = elem;this.succ = succ;

}

}

}

4-7

Example: SLL traversal

Method to traverse a SLL (in class SLL):

public void printFirstToLast () {// Print all elements in this SLL, in first-to-last order.

SLL.Node curr = this.first;while (curr != null) {

System.out.println(curr.element);curr = curr.succ;

}}

Animation:

ant bat catfirst ant bat catfirst

curr

ant bat catfirst

curr

ant bat catfirst

curr

ant bat catfirst

curr

4-8

ant bat catfirst ant bat catfirst

Example: SLL manipulation

Method to delete a SLL’s first node (in class SLL):

public void deleteFirst () {// Delete this SLL’s first node (assuming length > 0).

this.first = this.first.succ;}

Animation:

4-9

Insertion

Problem: Insert a new element at a given point in a linked-list.

Four cases to consider:

A. insertion in an empty linked-list;

B. insertion before the first node of a non-empty linked-list;

C. insertion after the last node of a non-empty linked-list;

D. insertion between nodes of a non-empty linked-list.

The insertion algorithm needs links to the new node’s successor and predecessor.

4-10

SLL insertion (1)

SLL insertion algorithm:

To insert elem at a given point in the SLL headed by first:

1. Make ins a link to a newly-created node with element elem

and successor null.2. If the insertion point is before the first node:

2.1. Set ins’s successor to first.2.2. Set first to ins.

3. Else, if the insertion point is after the node pred:3.1. Set ins’s successor to pred’s successor.3.2. Set pred’s successor to ins.

4. Terminate.

cases A, B

cases C, D

4-11

bat catfirst

To insert elem at a given point in the SLL headed by first:1. Make ins a link to a newly-created node with element elem and successor null.

2. If the insertion point is before the first node:2.1. Set ins’s successor to first.2.2. Set first to ins.


4. Terminate.

elem antant

bat catfirst

ins




4. Terminate.

elem antant

bat catfirst

ins




4. Terminate.

elem antant

bat catfirst

ins

To insert elem at a given point in the SLL headed by first:1. Make ins a link to a newly-created node with elementelem and successor null.



4. Terminate.

elem antant

bat catfirst

To insert elem at a given point in the SLL headed by first:1. Make ins a link to a newly-created node with elementelem and successor null.



4. Terminate.

elem ant

SLL insertion (2)

Animation (insertion before first node):

4-12

dog foxfirst




4. Terminate.

pred eelelem

dog foxfirst




4. Terminate.

eelpred ins eelelem

dog foxfirst




4. Terminate.

eelpred ins eelelem

dog foxfirst




4. Terminate.

eelpred ins eelelem

dog foxfirst




4. Terminate.

eel eelelempred

SLL insertion (3)

Animation (insertion after intermediate node):

4-13

SLL insertion (4)

Implementation as a Java method (in class SLL):

public void insert (Object elem,SLL.Node pred) {

// Insert elem at a given point in this SLL, either after the

// node pred, or before the first node if pred is null.SLL.Node ins = new SLL.Node(elem,

null);if (pred == null) {

ins.succ = this.first;this.first = ins;

} else {ins.succ = pred.succ;pred.succ = ins;

}}

4-14

Deletion

Problem: Delete a given node from a linked-list.

Four cases to consider:

A. deletion of a singleton node;

B. deletion of the first (but not last) node;

C. deletion of the last (but not first) node;

D. deletion of an intermediate node.

The deletion algorithm needs links to the deleted node’s successor and predecessor.

4-15

SLL deletion (1)

SLL deletion algorithm:

To delete node del from the SLL headed by first:

1. Let succ be del’s successor.2. If del = first:

2.1. Set first to succ.3. Else:

3.1. Let pred be del’s predecessor.3.2. Set pred’s successor to succ.

4. Terminate.

cases A, B

cases C, D

But there is no link from node del to its predecessor, so step 3.1 can access del’s predecessor only by following links from first!

4-16

To delete node del from the SLL headed by first:1. Let succ be del’s successor.2. If del = first:



4. Terminate.

ant bat catfirst

del




4. Terminate.

ant bat catfirst

del succ




4. Terminate.

ant bat catfirst

del succ




4. Terminate.

ant bat catfirst

garbage

SLL deletion (2)

Animation (deleting the first node):

4-17




4. Terminate.

dog eel foxfirst

del




4. Terminate.

dog eel foxfirst

del succ




4. Terminate.

dog eel foxfirst

del succpred




4. Terminate.

dog eel foxfirst

del succpred




4. Terminate.

dog eel foxfirst

garbage

SLL deletion (3)

Animation (deleting an intermediate node):

4-18

SLL deletion (4)

Analysis:

Let n be the SLL’s length.

Step 3.1 must visit all nodes from the first node to the deleted node’s predecessor. There are between 0 and n–1 such nodes.

Average no. of nodes visited = (n – 1)/2

Time complexity is O(n).

4-19

SLL deletion (5)


public void delete (SLL.Node del) {// Delete node del from this SLL.

SLL.Node succ = del.succ;if (del == this.first) {

this.first = succ;} else {

SLL.Node pred = this.first;while (pred.succ != del)

pred = pred.succ;pred.succ = succ;

}}

4-20

Searching

Problem: Search for a given target value in a linked-list.

Idea: Follow links from the first node to the last node, terminating when we find a node whose element matches the target value.

4-21

SLL searching (1)

Unsorted SLL linear search algorithm:

To find which (if any) node of the SLL headed by first contains an element equal to target:

1. For each node curr in the SLL headed by first, repeat:1.1. If target is equal to curr’s element, terminate with

answer curr.2. Terminate yielding none.

4-22

SLL searching (2)

Analysis (counting comparisons):

Let n be the SLL’s length.

If the search is successful:

Average no. of comparisons = (n + 1)/2 n/2

If the search is unsuccessful:

No. of comparisons = n

In either case, time complexity is O(n).

4-23

SLL searching (3)


public SLL.Node search (Object target) {// Find which (if any) node of this SLL contains an // element equal to target. Return a link to the // matching node (or null if there is none).

SLL.Node curr = this.first;while (curr != null) {

if (target.equals(curr.element))return curr;

curr = curr.succ;}return null;

}

4-24

Other SLL algorithms

Some other array algorithms can easily be adapted to SLLs:

– merging

– selection sort

– merge-sort

But not:

– binary search.

4-25

Doubly-linked-lists (1)

A doubly-linked-list (DLL) consists of a header together with a sequence of nodes connected by links in both direction:

– Each DLL node contains a single element, plus a link to the node’s successor (or null), plus a link to the node’s predecessor (or null).

– The DLL header contains links to the DLL’s first and last nodes (or null if the DLL is empty).

pig dog ratcat

dogempty DLL

4-26


Java class implementing DLLs:

public class DLL {// Each DLL object is the header of a // doubly-linked-list.

private DLL.Node first, last;

public DLL () {// Construct an empty DLL.this.first = null;this.last = null;

}

… methods (to follow)

4-27


Java class implementing DLLs (continued):////////// Inner class //////////

private static class Node {// Each DLL.Node object is a node of a // doubly-linked-list.

protected Object element;protected Node pred, succ;

public Node (Object elem,Node pred, Node succ) {

this.element = elem;this.pred = pred; this.succ =

succ;}

}

}

4-28

Example: DLL traversal

Method to traverse a DLL (in class DLL):

public void printLastToFirst () {// Print all elements in this DLL, in last-to-first order.

DLL.Node curr = this.last;while (curr != null) {

System.out.println(curr.element);curr = curr.pred;

}}

Animation:

ant bat catfirstlast


curr


curr


curr


curr


4-29

Example: DLL manipulation (1)

Method to delete a DLL’s first node (in class DLL):

public void deleteFirst () {// Delete this DLL’s first node (assuming length > 0).

DLL.Node second = this.first.succ;second.pred = null;this.first = second;

}

Animation:


second


second


second


4-30

Example: DLL manipulation (2)

Method to delete a DLL’s last node (in class DLL):

public void deleteLast () {// Delete this DLL’s last node (assuming length > 0).

DLL.Node penult = this.last.pred;penult.succ = null;this.last = penult;

}

Animation:


penult


penult


penult


4-31

DLL insertion (1)

DLL insertion algorithm:

To insert elem at a given point in the DLL headed by (first, last):

1. Make ins a link to a newly-created node with element elem

and with predecessor and successor both null.2. If first = last = null:

2.1. Set first to ins.2.2. Set last to ins.

3. Else, if the insertion point is before node first:3.1. Set ins’s successor to first.3.2. Set first’s predecessor to ins.3.3. Set first to ins.

4. Else, if the insertion point is after node last:4.1. Set ins’s predecessor to last.4.2. Set last’s successor to ins.4.3. Set last to ins.

4-32

DLL insertion (2)

DLL insertion algorithm (continued):

5. Else, if the insertion point is after the node pred and before

the node succ:5.1. Set ins’s predecessor to pred.5.2. Set ins’s successor to succ.5.3. Set pred’s successor to ins.5.4. Set succ’s predecessor to ins.

6. Terminate.

4-33

DLL deletion

DLL deletion algorithm:

To delete node del from the DLL headed by (first, last):

1. Let succ be del’s successor, and let pred be del’s successor.

2. If del = first:2.1. Set first to succ.

3. Else:3.1. Set pred’s successor to succ.

4. If del = last:4.1. Set last to pred.

5. Else:5.1. Set succ’s predecessor to pred.

6. Terminate.

4-34

Comparison of SLL and DLL algorithms

Algorithm SLL DLL

Insertion O(1) O(1)

Deletion O(n) O(1)

Search O(n) O(n)

4 linked-list data structures linked-lists: singly-linked-lists, doubly-linked-lists insertion ...

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